[Proof.]
Without loss of generality, we prove when \(\mathcal {Z}=\mathbf {1}_{p(p-1)/2}\) , where \(\mathbf {1}_{p(p-1)/2}\)  is \(p(p-1)/2-\) vector with all entries being 1, since the proof is essentially the same for general \(\mathcal {Z}\) . The difference is only fixed zero entries of \(\Sigma _{\mathcal {Z}}\) . In the proof, we write \(\Sigma _{\mathcal {Z}}\)  as \(\Sigma \)  for simplicity suppressing dependency on \(\mathcal {Z}\) . We prove Lemma REF  using the techniques considered in the proof of Theorem 1 in [1]}.
